Random Walkers with Shrinking Steps in d-Dimensions and Their Long Term Memory
Abstract
We study, in d-dimensions, the random walker with geometrically shrinking step sizes at each hop. We emphasize the integrated quantities such as expectation values, cumulants and moments rather than a direct study of the probability distribution. We develop a 1/d expansion technique and study various correlations of the first step to the position as ti me goes to infinity. We also show and substantiate with a study of the cumulants that to order 1/d the system admits a continuum counterpart equation which can be obtained with a generalization of the ordinary technique to obtain the continuum limit. We also advocate that this continuum counterpart equation, which is nothing but the ordinary diffusion equation with a diffusion constant decaying exponentially in continuous time, captures all the qualitative aspects of t he discrete system and is often a good starting point for quantitative approximations.
Cite
@article{arxiv.cond-mat/0608059,
title = {Random Walkers with Shrinking Steps in d-Dimensions and Their Long Term Memory},
author = {Tonguc Rador},
journal= {arXiv preprint arXiv:cond-mat/0608059},
year = {2009}
}